MathDB
Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 2
2016 Nigerian Senior MO Round 2
2016 Nigerian Senior MO Round 2
Part of
Nigerian Senior Mathematics Olympiad Round 2
Subcontests
(10)
Problem 3
1
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Sums and probability
The integers
1
,
2
,
…
,
9
1, 2, \dots , 9
1
,
2
,
…
,
9
are written on individual slips of paper and all are put into a bag. Ade chooses a slip at random, notes the integer on it, and replaces it in the bag. Bala then picks a slip at random and notes the integer written on it. Chioma then adds up Ade's and Bala's numbers. What is the probability that the unit's digit of this sum is less that
5
5
5
?
Problem 8
1
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Polynomials and manipulating roots
If
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are the solutions of the equation
x
4
−
k
x
−
15
=
0
x^4-kx-15=0
x
4
−
k
x
−
15
=
0
, find the equation whose solutions are
a
+
b
+
c
d
2
,
a
+
b
+
d
c
2
,
a
+
c
+
d
b
2
,
b
+
c
+
d
a
2
\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}
d
2
a
+
b
+
c
,
c
2
a
+
b
+
d
,
b
2
a
+
c
+
d
,
a
2
b
+
c
+
d
.
Problem 5
1
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Pyramid and combinatorics
A solid pyramid
T
A
B
C
D
TABCD
T
A
BC
D
, with a quadrilateral base
A
B
C
D
ABCD
A
BC
D
is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, what is the number of ways to colour the pyramid?
Problem 9
1
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Parallelograms and ratio of areas
A
B
C
D
ABCD
A
BC
D
is a parallelogram, line
D
F
DF
D
F
is drawn bisecting
B
C
BC
BC
at
E
E
E
and meeting
A
B
AB
A
B
(extended) at
F
F
F
from vertex
C
C
C
. Line
C
H
CH
C
H
is drawn bisecting side
A
D
AD
A
D
at
G
G
G
and meeting
A
B
AB
A
B
(extended) at
H
H
H
. Lines
D
F
DF
D
F
and
C
H
CH
C
H
intersect at
I
I
I
. If the area of parallelogram
A
B
C
D
ABCD
A
BC
D
is
x
x
x
, find the area of triangle
H
F
I
HFI
H
F
I
in terms of
x
x
x
.
Problem 7
1
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Irrationals and powers
Prove that
(
2
+
3
)
2
n
+
(
2
−
3
)
2
n
(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}
(
2
+
3
)
2
n
+
(
2
−
3
)
2
n
is an even integer and that
(
2
+
3
)
2
n
−
(
2
−
3
)
2
n
=
w
3
(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}
(
2
+
3
)
2
n
−
(
2
−
3
)
2
n
=
w
3
for some positive integer
w
w
w
, for all integers
n
≥
1
n \geq 1
n
≥
1
.
Problem 10
1
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Manpulating Logarithms
Positive numbers
x
x
x
and
y
y
y
satisfy
x
y
=
2
15
xy=2^{15}
x
y
=
2
15
and
log
2
x
⋅
log
2
y
=
60
\log_2{x} \cdot \log_2{y} = 60
lo
g
2
x
⋅
lo
g
2
y
=
60
. Find
(
log
2
x
)
3
+
(
log
2
y
)
3
3
\sqrt[3]{(\log_2{x})^3+(\log_2{y})^3}
3
(
lo
g
2
x
)
3
+
(
lo
g
2
y
)
3
Problem 6
1
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Simple inequality
Given that
a
,
b
,
c
,
d
∈
R
a, b, c, d \in \mathbb{R}
a
,
b
,
c
,
d
∈
R
, prove that
(
a
b
+
c
d
)
2
≤
(
a
2
+
c
2
)
(
b
2
+
d
2
)
(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)
(
ab
+
c
d
)
2
≤
(
a
2
+
c
2
)
(
b
2
+
d
2
)
.
Problem 4
1
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Real numbers and roots
Find the real number satisfying
x
=
1
+
1
+
1
+
x
x=\sqrt{1+\sqrt{1+\sqrt{1+x}}}
x
=
1
+
1
+
1
+
x
.
Problem 2
1
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Ratios and trigonometry
P
Q
PQ
PQ
is a diameter of a circle.
P
R
PR
PR
and
Q
S
QS
QS
are chords with intersection at
T
T
T
. If
∠
P
T
Q
=
θ
\angle PTQ= \theta
∠
PTQ
=
θ
, determine the ratio of the area of
△
Q
T
P
\triangle QTP
△
QTP
to the area of
△
S
R
T
\triangle SRT
△
SRT
(i.e. area of
△
Q
T
P
\triangle QTP
△
QTP
/area of
△
S
R
T
\triangle SRT
△
SRT
) in terms of trigonometric functions of
θ
\theta
θ
Problem 1
1
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Complex numbers and symmetry
Let
a
,
b
,
c
,
x
,
y
a, b, c, x, y
a
,
b
,
c
,
x
,
y
and
z
z
z
be complex numbers such that
a
=
b
+
c
x
−
2
,
b
=
c
+
a
y
−
2
,
c
=
a
+
b
z
−
2
a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}
a
=
x
−
2
b
+
c
,
b
=
y
−
2
c
+
a
,
c
=
z
−
2
a
+
b
. If
x
y
+
y
z
+
z
x
=
1000
xy+yz+zx=1000
x
y
+
yz
+
z
x
=
1000
and
x
+
y
+
z
=
2016
x+y+z=2016
x
+
y
+
z
=
2016
, find the value of
x
y
z
xyz
x
yz
.